Advanced data assimilation methods with evolving forecast error covariance

Four-dimensional variational analysis(4D-Var)

Shu-Chih Yang (with EK)

Find the optimal analysis

• Least squares approachFind the optimal weights to minimize the analysis error covariance

€

Ta =σ 2

2

σ 12 + σ 2

2

⎛

⎝ ⎜

⎞

⎠ ⎟T1 +

σ 12

σ 12 + σ 2

2

⎛

⎝ ⎜

⎞

⎠ ⎟T2

€

J(T) =1

2

(T − T1)2

σ 12

+(T − T2)2

σ 22

⎡

⎣ ⎢

⎤

⎦ ⎥ ,

∂J

∂T= 0 for T = Ta

• Variational approachFind the analysis that will minimize a cost function, measuring its distance to the background and to the observation

€

T1 = Tt + ε1

T2 = Tt + ε2

(forecast)

(observation)Best estimate the true value

Both methods give the same Ta !

J(x)= (x-xb)TB-1(x-xb) + [yo-H(x)]TR-1[yo-H(x)]

J(xa)=0 at J(xa)=Jmin

3D-Var

Distance to forecast (Jb) Distance to observations (Jo)

€

1

2

€

1

2

How do we find an optimum analysis of a 3-D field of model variable xa, given a background field, xb, and a set of observations, yo?

find the solution in 3D-Var

Directly set J(xa)=0 and solve

(B-1+HTR-1H)(xa-xb)=HTR-1[yo-H(xb)] (Eq. 5.5.9)

Usually solved as

(I+ B HTR-1H)(xa-xb)= B HTR-1[yo-H(xb)]

€

δJ ≈∂J

∂x

⎡ ⎣ ⎢

⎤ ⎦ ⎥

T

⋅δx; ∇J =∂J

∂x

Minimize the cost function, J(x)

A descent algorithm is used to find the minimum of the cost function.

This requires the gradient of the cost function, J.

-J(x)

-J(x+1)

J(x)

J(x+1)

min

Ex: “steepest descent” method

4D-Var

J(x(t0))= [x(t0)-xb(t0)]TB0

-1[x(t0)-xb(t0)]+ [yoi-H(xi)]

TRi-1[yo

i-H(xi)]

Need to define J(x(t0)) in order to minimize J(x(t0))€

1

2

€

1

2 i= 0

i= N

∑

Find the initial condition such that its forecast best fits the observations within the assimilation interval

assimilation windowt0

tnti

yo

yo

yo

yo

previous forecast

xb corrected forecast

xa

J(x) is generalized to include observations at different times.

Jb =12[x(t0 )−xb(t0 )]

T B−1[x(t0 )−xb(t0 )]

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∂Jb

∂x(t0)= B−1[x(t0) − xb (t0)]

Given a symmetric matrix A, and

a function , the gradient is give by

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J =1

2xTAx

€

∂J

∂x= Ax

J =J b + J o,∂J

∂x(t0 )=

∂J b

∂x(t0 )+

∂J o

∂x(t0 )

First, let’s consider Jb(x(t0))

Separate J(x(t0)) into “background” and “observation” terms

∂(H (xi ) − yio )

∂x0

= ∂H

∂xi

∂M i

∂x0

= HiL(t0 , ti ) = HiLi-1Li-2 L L0

Jo is more complicated, because it involves the integration of the model:

Jo =12

[H(xi )−yio]Ri

-1[H(xi )−yio]

i=0

N

∑

∂Jo

∂x(t0 )

⎡

⎣⎢

⎤

⎦⎥= LT (t0 ,ti )Hi

T R i-1[H (xi ) − yi

o ]i=0

N

∑weighted increment at observation time, ti, in model coordinates

If J = yTAy and y = y(x), then ,

where is a matrix.€

∂J

∂x=

∂y

∂x

⎡ ⎣ ⎢

⎤ ⎦ ⎥

T

Ax

€

∂y

∂x

⎡ ⎣ ⎢

⎤ ⎦ ⎥k,l

=∂yk

∂x l

xi=Mi[x(ti-1)]

[HiLi-1Li-2 ...L0 ]T =L 0T L L i−2

T L i−1T H i

T =LT (ti ,t0 )H iT

Adjoint model integrates increment backwards to t0

€

d i = H iTRi

−1[H(x i) − y io]

t0 t1 t3t2 t4

€

d 4

€

d 3

€

d 2

€

d 1

€

d 0 + L0T (d 1 + L1

T (d 2 + L2T (d 3 + L3

T d 4 )))

€

B0−1[x(t0) − xb (t0)]

+Jo/x0

Jb/x0

€

d 0

• In each iteration, J is used to determine the direction to search the Jmin.

• 4D-Var provides the best estimation of the analysis state and error covariance is evolved implicitly.

Simple example:Use the adjoint model to integrate backward in time

Start from the end!

Jo

t0tnti

assimilation window

yo

yo

yo

yo

previous forecast

corrected forecastxb

Jb

Jo

Jo

Jo

xa

3D-V

ar

1. 4D-Var assumes a perfect model. It will give the same credence to older observations as to newer observations.• algorithm modified by

Derber (1989)2. Background error covariance

is time-independent in 3D-Var, but evolves implicitly in 4D-Var.

3. In 4D-Var, the adjoint model is required to compute J.

Figure from http://www.ecmwf.int/

3D-Var vs. 4D-Var

With this form, it is possible to choose a “simplification operator, S” to solve the cost function in a low dimension space (change the control variable).Now, δw=Sδx and minimize J(δw)

where and

Practical implementation: use the incremental form

The choice of the simplification operator• Lower resolution• Simplification of physical process

€

J(δx0) =1

2(δx0)

T B0−1δx0 +

1

2[H iL(t0, ti)δx0 − di

o]T R−1

i= 0

N

∑ [H iL(t0, ti)δx0 − dio]

€

δx = x − xb

€

d = yo − H(x)

Example of using simplification operator

Both TLM and ADJ use a low resolution and also simplified physics due to the limitation of the computational cost.

Example with the Lorenz 3-variable model

• The background state is needed in both L and LT (need to save the model trajectory)

• In a complex NWP model, it is impossible to write explicitly this matrix form

€

dx1

dt= −px1 + px2

dx2

dt= rx1 − x1x3 − x2

dx3

dt= x1x2 − bx3

Nonlinear model x=[x1,x2,x3]

Tangent linear model δx=[δx1, δx2, δx3]

Adjoint model δx*=[δx*

1, δx*2, δx*

3]

€

L =∂M

∂x=

∂M

∂x i

=

−p p 0

r − x3 −1 −x1

x2 x1 −b

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

x3 x1

x2 x1

€

LT =∂M

∂x i

⎡

⎣ ⎢

⎤

⎦ ⎥

T

=

−p r − x3 x2

p −1 x1

0 −x1 −b

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥x1

x1

x2x3

In tangent linear model

forward in time

In adjoint model

backward in time

Example of tangent linear and adjoint codes

€

δx 3(t + Δt) −δx3(t)

Δt= x2(t)δx1(t) + x1(t)δx2(t) − bδx3(t)

δx 3(t + Δt) = δx3(t) + [x2(t)δx1(t) + x1(t)δx2(t) − bδx3(t)]Δt

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δx3*(t) = δx3

*(t) + (1− bΔt)δx3*(t + Δt)

δx2*(t) = δx2

*(t) + (x1(t)Δt)δx3*(t + Δt)

δx1*(t) = δx1

*(t) + (x2(t)Δt)δx1*(t + Δt)

δx1*(t + Δt) = 0

* Try an example in Appendix B (B.1.15)

use forward scheme to integrate in time

RMS error of 3D-Var and 4D-Var in Lorenz model

Experiments: DA cycle and observations: 8t, R=2*I 4D-Var assimilation window: 24t

observation error

3DVar

4DVar

4D-Var in the Lorenz model (Kalnay et al., 2005)

Impact of the window length

Win=8 16 24 32 40 48 56 64 72

Fixed window 0.59 0.59 0.47 0.43 0.62 0.95 0.96 0.91 0.98

Start with short window

0.59 0.51 0.47 0.43 0.42 0.39 0.44 0.38 0.43

• Lengthening the assimilation window reduces the RMS analysis error up 32 steps.

• For the long windows, error increases because the cost function has multiple minima.

• This problem can be overcome by the quasi-static variational assimilation approach (Pires et al, 1996), which needs to start from a shorter window and progressively increase the length of the window.

Schematic of multiple minima and increasing window size (Pires et al, 1996)

failed

1 2final

……

Jmin1

Jmin2

J(x)

Dependence of the analysis error on B0

Dependence of the analysis error on the B0

•Since the forecast state from 4D-Var will be more accurate than 3D-Var, the amplitude of B should be smaller than the one used in 3D-Var.• Using a covariance proportional to B3D-Var and tuning its amplitude is a good strategy to estimate B.

Win=8 B= B3D-Var 50%

B3D-Var

40%

B3D-Var

30%

B3D-Var

20%

B3D-Var

10%

B3D-Var

5%

B3D-Var

RMSE 0.78 0.59 0.53 0.52 0.50 0.51 0.65 >2.5